COMPLEX NUMBERS TUTORIAL

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· Section 4: Complex Numbers -

Polar and Exponential Forms

· Objective: Present definitions and examples using polar and exponential forms of complex numbers. Our goal is to establish the relationships indicated in the following table:

x + yj =	r (cos q + jsin q )	= re^j^q
Rectangular form	Polar form	Exponential form
	q can be in degrees OR radians	q MUST be in radians

4.1 Polar Form:

· Just as a point in the plane with rectangular coordinates (a,b) can be written in polar coordinates, we can introduce polar form for complex numbers.

As noted in Section 3, we can think of complex numbers as vectors.

We find the real (horizontal) and imaginary (vertical) components in terms of r (the length of the vector) and q (the angle measured counterclockwise from the positive real axis).

From Pythagoras, we have: r² = x² + y² and basic trigonometry gives us:

x = r cos q

y = r sin q

Multiplying the last expression throughout by j gives us:

yj = jr sin q

So we can write the polar form of a complex number as:

*4.2 We can also represent complex numbers in exponential form using Euler’s formula.

Polar and Exponential Forms

The standard form of a complex number is

but this can be shown to be equivalent to the form

which is called the exponential form of a complex number. This form is equivalent to the polar form as well. The equivalence can be shown by using Euler’s formula,

Proof:

Note: A is equivalent to our r notation above.

Summary of Exponential Form:

The complex number a+jb can be written:

re^jq, where

· r is called the modulus and denotes the magnitude

· q is called the argument of a+jb and denotes the angle measured counterclockwise from the positive real axis.

· Example:

In the example below we convert from rectangular to polar and exponential form.

Example:  Convert the complex number  from rectangular to polar form.

Solution:

Since  and  lies in the first quadrant, choose

The complex number is written as:

* In the next example below we convert from polar to rectangular form.

· Summary: Our observations from all of the above are summarized in the following table referenced in the objectives.

x + yj =	r (cos q + jsin q) = r cos q = rÐ q	= re^j^q
rectangular form	polar form	exponential form
	q can be in degrees OR radians	q MUST be in radians

*Note: r cos q and r Ð q are equivalent forms for complex numbers; they are used throughout electrical engineering. See the next section for more detail.

Exercises

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